What Is the Matching Law?
The matching law, first described by Richard Herrnstein in 1961, states that the relative rate of responding on a particular alternative matches the relative rate of reinforcement earned from that alternative. In simpler terms, organisms distribute their behavior in proportion to the reinforcement available for each option.
Herrnstein's original research involved pigeons pecking two keys on concurrent variable interval (VI) schedules. He found that the proportion of pecks on one key closely matched the proportion of reinforcers delivered by that key. This relationship is expressed mathematically as:
B1 / (B1 + B2) = R1 / (R1 + R2)
Where B represents the rate of behavior and R represents the rate of reinforcement for each alternative (1 and 2).
Beyond the Basic Equation
The generalized matching law, developed by Baum in 1974, expanded Herrnstein's work to account for deviations from strict matching. The generalized equation introduces two parameters: bias and sensitivity.
Bias
Bias refers to a consistent preference for one alternative over another that is not explained by differences in reinforcement rate. Bias can result from factors like response effort, quality of reinforcement, or the organism's learning history. In a clinical context, a child might prefer one activity over another even when the reinforcement rates are equal, simply because one requires less effort.
Sensitivity
Sensitivity describes how closely the organism's behavior tracks changes in reinforcement ratios. Perfect matching yields a sensitivity of 1.0. Undermatching (sensitivity less than 1.0) occurs when behavior is less differentiated than reinforcement ratios would predict, while overmatching (sensitivity greater than 1.0) means behavior is more differentiated than expected.
Clinical Relevance for BCBAs
The matching law has profound implications for understanding and treating problem behavior in clinical settings.
Understanding Problem Behavior as a Choice
When a client engages in problem behavior, they are choosing that behavior from among the available alternatives. The matching law tells us that this choice reflects the relative reinforcement available for problem behavior compared to appropriate alternatives. If a child's tantrum produces immediate, high-quality attention (reinforcement) while appropriate requests are sometimes ignored, the matching law predicts the child will allocate more behavior toward tantrums.
Designing Competing Contingencies
Functional communication training (FCT) is one of the most well-supported interventions in ABA, and its effectiveness can be understood through the matching law. FCT works by providing a reinforced alternative response that competes with problem behavior. The matching law predicts that the client will shift responding toward the alternative when the reinforcement for the alternative exceeds the reinforcement for the problem behavior in rate, quality, immediacy, or effort.
To maximize the effectiveness of FCT or any differential reinforcement procedure, clinicians should consider all the dimensions that influence choice:
- Rate of reinforcement: How often is each behavior reinforced?
- Quality of reinforcement: Are the reinforcers equally preferred?
- Immediacy: How quickly does reinforcement follow each behavior?
- Response effort: How difficult is each behavior to perform?
Concurrent Schedules in Everyday Life
Clients are always operating under concurrent schedules of reinforcement. In a classroom, a student can choose to raise their hand, call out, sit quietly, or leave their seat. Each of these options has its own reinforcement history and current contingencies. Understanding behavior as allocation across alternatives, rather than as isolated responses, gives clinicians a more accurate and useful framework for intervention.
Practical Applications
Increasing Appropriate Behavior
The matching law suggests that to increase appropriate behavior, clinicians should not only reinforce the target behavior but also reduce the reinforcement available for competing responses. This is why extinction is often a component of effective interventions. Reinforcing appropriate behavior while simultaneously reducing reinforcement for problem behavior shifts the reinforcement ratio in favor of the desired response.
Evaluating Intervention Failures
When an intervention is not producing expected results, the matching law provides a diagnostic framework. If appropriate behavior is not increasing despite reinforcement, consider whether problem behavior is still producing reinforcement that is more immediate, higher quality, or more frequent. The issue may not be that the reinforcement for the target behavior is insufficient on its own, but that it cannot compete with the reinforcement maintaining the problem behavior.
Preference Assessments and the Matching Law
Conducting thorough preference assessments ensures that the reinforcers used in intervention are potent enough to shift behavior allocation. The matching law reminds us that it is the relative, not the absolute, value of reinforcement that drives behavior allocation.
Exam Preparation Tips
For the BCBA exam, be prepared to recognize scenarios that illustrate the matching law. Know the difference between strict matching, undermatching, and overmatching. Understand how concurrent schedules of reinforcement relate to clinical decision-making, and be able to apply the matching law to explain why a client might prefer one behavior over another. Questions may present a scenario and ask you which variable change would most effectively shift behavior allocation.